- Lebesgue integration, measure theory, convergence theorems, the Riesz representation theorem, Fubini's theorem, complex measures
*L^p*-spaces, density of continuous functions, Hilbert space, weak and strong topologies, integral operators.- Inequalities.
- Bounded linear operators and functionals. Hahn-Banach theorem, open-mapping theorem, closed graph theorem, uniform boundedness principle.
- Schwartz space, introduction to distributions, Fourier transforms
on the circle and the line (Schwartz space and
*L^2*). - Spectral theorem for bounded normal operators.

__Textbooks__:

Kolmogorov, A.N. and Fomin, S.V., "Introductory Real Analysis", 1975.

There will be a second text, to be chosen later.

__References__:

Folland, G.B., "Real Analysis: Modern Techniques and their Applications"

Lieb, E.H. and Loss, M., "Analysis"

Royden, H.L., "Real Analysis"

Rudin, W., "Real and Complex Analysis"

Rudin, W., "Functional Analysis"

Taylor, A.E., "Introduction to Functional Analysis"

Torchinsky, A., "Real Variables"

Yoshida, K., "Functional Analysis"

Zimmer, R.J., "Essential Results of Functional Analysis"

I have put all but one of the above books on reserve at the Math-Stat
library in the basement of Sidney Smith. (The university doesn't currently have Zimmer's book. It's
a lovely book and costs only $29.54 at
amazon.ca) You cannot check the
books out --- you have to use them while in the library. The
library's hours are Monday to Friday, 9:00-5:00. Please let me know
if the restrictions on their use is a real problem for you.
Looking to buy used books? I've had good luck with
abebooks.com.

Your course mark will be based on homework (worth 20%), three term
exams (worth 15% each), and one final exam (worth 35%).

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Your first homework assignment, due Wednesday September 18.
The solutions to eight of the warm-up exercises were penned by
by Steven Sloot and Jacob Sone. (Thanks!!)
Here they are, please let me know if you find any problems with them.

Your second homework assignment, due Friday September 27. Here are the solutions. The solutions to eleven of the warm-up exercises were penned by by Sajiya Jalil and Carina Siu. (Thanks!!) Here they are, please let me know if you find any problems with them.

Your third homework assignment, due Friday October 4. Here are the solutions. Note: there's a mistake in the solution to problem 5a. Because the matrices are _real_ valued, we cannot diagonalize when the eigenvalues are complex. If the eigenvalues are a +/- i b, then the reduced form of the matrix will be A_11 = a, A_12, = b, A_21 = -b, A_22 = a.

See the Tietze Extension Theorem in Action! Just save the file tietze_extension.m in your home directory, start up matlab (in your home directory) and type "tietze_extension" at the prompt. If you're a hacker, open it up and change the function f and the number of approximants and whatever...

Your fourth homework assignment, due Wednesday October 16. Here are the solutions.

Google is our friend. Here are some topology notes from Kazuo Yokoyama of Sophia University in Japan. Here are some topology notes from Michael Van Opstall of the University of Washington. (Please let me know if you find mistakes in either document.)

See the Stone Weierstrauss approximants in Action! Just save the file stone_weierstrauss.m in your home directory, start up matlab (in your home directory) and type "stone_weierstrauss" at the prompt. If you're a hacker, open it up and change the function f and the number of approximants and whatever...

Your fifth homework assignment, due Friday October 25, is from Kolmogorov and Fomin: problem 6 on page 128 and problems 1-4 on page 137. Here are the solutions.

Your first term test was in class on Monday October 28. It is worth 15% of your course mark. Here are the solutions.

Your sixth homework assignment, due Friday, November 15. The solutions to problems 3-5 are at the end of the solutions to the second term test.

Google is our friend. Here are some notes on topological vector spaces from Paul Garrett of the University of Minnesota.

Also, see Rudin's "Functional Analysis" pages 1-13.

There will be no class or office hour on Friday November 15. I'll schedule a make-up class later. You can hand the homework in on Monday November 18. On November 14, my office hour will be 1-2 rather than the usual time of 3-4.

Here is an example of a space X which is isometric to its second dual X^**, but which is not isometric with X^** when you use the natural mapping from X to X^**. The example was discovered by R.D.James and published in 1951.

Your seventh homework assignment, due Monday, December 2, is from Kolmogorov and Fomin: page 171 #12, page 183 #8, page 194 #7 and #9, page 205 #2 and #3.

There will be no class on Wednesday December 4. I'll schedule a make-up class later. There will be no office hours on Monday December 2 or Thursday December 5.

Your second term test was in class on Wednesday January 8. It is worth 15% of your course mark. Here are the solutions.

Your eighth homework assignment, due Wednesday, January 29.

Remember! No class on Wednesday January 22!

Your ninth homework assignment, due Friday, February 14. Please put it into Ching-Nam Hung's mailbox by 4 pm. Here are the solutions.

The book for the measure theory and integration portion of the course will be Daniel W. Stroock's "A Concise Introduction to the Theory of Integration". It is in stock in the UofT bookstore.

Here are some practice problems on compact operators and spectral theory for you to work in to help you prepare for the term exam.

The third term test was in class on Wednesday March 5. It covered distributions, linear operators, and spectral theory.

Ching-Nam Hung will have twenty hours of office hours the week before the exam. His office is SS 4052. His office hours will be: Friday Apr 25 1-5pm, Monday Apr 28 1-5pm, Tuesday Apr 29 2:30-6:30pm, Wednesday Apr 30 9am-1pm, and Thursday May 1 9am-1pm.

I will have office hours on Monday April 28 1pm-4pm.

Here are some old UofT comprehensive exams.

Here are some old UIUC comprehensive exams.

Here are some review notes, courtesy of David Rose of UIUC: set 1, set 2, set 3, and set 4.